Containers- Minimum area

An Investigation

ContainersConsider the container

It is required to design a box which satisfies the following requirements:-

a) The box is to hold 2000cm3 (2 litres)

b) The length of the box is to be two times the

breadth of the box.

c) The amount of cardboard needed to make the

box is to be a minimum.

x

2x

h

Steps:- 1. Use the fact that V = LBH to show that 2 Show that the total area of cardboard

needed is

3. Put into Y1 and use the TABLE

function on the graphic calculator to obtain the solution.

2

1000

xh

246000

xx

A

246000

xx

Table 1 Complete the table shown

to show your results.

X Area (cm2)

1 6004

2 3016

3

4

5

6

7

8

9

10

11

12

You can see from the table that the minimum is somewhere between x = 8 and x = 10.

So we now form another table with x going up in steps of 0.2 say.

On your calculator set TblStart to 8 and ∆Tbl to 0.2 and then complete the Table 2 on the next slide.

Table 2

Why have we

stopped this

Table at 9.6?

X Area (cm2)

8 1006

8.2 1000.7

8.4

8.6

8.8

9

9.2

9.4

9.6

Make up a third table starting at ? and going up in steps of 0.1 to obtain the final value for x which will give the minimum area of cardboard.

Now that you have this value of x, draw a box like the one below and mark on the box the dimensions which the box would have for this minimum area.

Note

A company using this box for packaging goods, would produce millions of these boxes and the cost of producing them would be a minimum because each box uses the smallest amount of cardboard and satisfies the other conditions stated initially in the problem.

Clean ItWashing Powder

Container Examples(Using the graphic calculator Table function)

1. The container is to hold a volume of 800cm3.

The length is to be five times the breadth.

The area of cardboard needed to make the container is to be a minimum.

Find x, then draw the container.

h

5x

x

Ex2. Design a container to satisfy the following requirements.

The container is to hold a volume of 1200cm3.

The container has a square base.

The area of cardboard needed to make the container is to be a minimum.

h

xx

Ex3. Design the container below

The container is to hold a volume of 1500cm3.

The length is to be two times the breadth.

The area of cardboard needed to make the container is to be a minimum.

h

x

2x

Ex.4This container is to hold a volume of 1600cm3.

The length is to be three times the breadth.

The area of cardboard needed to make the container is to be a minimum.

h

3x

x